This question may be very trivial in this area, but I am a beginner to this and I stuck here. Could anyone please help me here!
Let $\Gamma$ be a group whose identity is $e$.
Let $X$ be a set and $∗:\Gamma×S\rightarrow S$ be a group action.
Let $[x]$ is the orbit (equivalence class) of $x\in X$ under the group action $\Gamma$.
Now I want to define addition in the quotient space $X/\Gamma$.
Here is my attempt:
Take $x'\in [x]$ and $y'\in [y]$ where $[x]\cap[y]=\phi$.
Then to define sum uniquely, I need to show $x+x'\sim_G y+y'$.
To show the above: Since $x'\in [x]$ and $y'\in [y]$ $$\exists g_1\in \Gamma: x'=g_1*x, \;\;\exists g_2\in \Gamma: y'=g_2*y.$$
Now $$x'+y'=g_1*x+g_2*y$$ Here I stuck: since $g_1$ and $g_2$ may be different element of $\Gamma$, then how to show $x'+y'\sim_\Gamma x+y$?